Tuesday, 20 December 2016

Mathematical Modeling Of Excess Fats On Blood Flow In Human Arteries

The study of blood flow has become more important and necessary coupling with the ever-increasing cardiac related problems which are usually as a result of one form of abnormality on blood flow or another. Under normal condition, the amount of inlet blood flow into an artery equals that which leaves i.e. the outlet blood. As a result of fats buildup on the walls of an artery, the diameter of the reduces hence affects the flow profile.  In this research work, we consider the case of excess fats as an external factor affecting the blood flow. The continuity equation, the Navier-Stokes equations for a cylindrical tube as well as Poissuile equation is used in formulating the model. MATLAB software is used in the analytical solution of the problem. The result shows that the flow profile is affected by the pressure variation, the radius of the artery and as such, the flow profile in a healthy artery differs from that with an affected artery. From the analysis we can detect the existence of excess fat through blood flow velocity within the artery.        

Title page

1.0       Introduction  
1.1       Problem Statement
1.2       Aim and Objectives
1.3       Significance of Study
1.4.      Scope and Limitations
1.5.0Definition of physiological terms used
1.5.3Types of fats     
1.5.4Cardiovascular System     1.5.5Function of the heart  
1.5.6Blood Vessels 
1.6.1Definition of fluid terms 
1.6.2Review of Partial Differential Equations
1.6.5Methods of forming PDE’S
1.6.6 Classification of PDE’S

2.0 Introduction
2.1 Review of Related Literatures 
3.0 Problem Formation  
3.1 Theorem(Poissuille law of flow)
3.2 Model Assumptions   
3.3 Application of  Model Assumption in Artery
Figure 4.1 mean velocity against length at no stenosis
Figure 4.2 Geometry of a healthy artery
Figure 4.3 mean velocity against length at difference pressures
Figure 4.4 geometry of a slightly affected artery due to fats buildup
Figure 4.5 mean velocity against length for change in stenosis at the same point
Figure 4.6 geometry of a severely affected artery due to fats buildup
Table 4.1 numeric data for R, A, and V respectively
5.0 Summary
5.1 Conclusion
 Appendix 1: Nomenclature
Appendix 2: Log Sheet



One of the leading causes of deaths in the world today is due to heart diseases and the most common heard names are Ischemia, Atherosclerosis and Angina pectoris. Ischemia is the deficiency of oxygen in a part of the body, usually temporary. It is mostly due to a constriction (stenoses) or obstruction in the blood vessel supplying that part. Atherosclerosis on the other hand is a type of arteriosclerosis which comes from the Greek words Arthero (meaning gruel or paste) and sclerosis (hardness). It involves deposits of fatty substances, cholesterol, cellular waste products etc, in the inner lining of an artery. Pralhad and Schultz, (2004).

The buildup of either of these substances in the walls of the artery forms what we call plaque. Plaque may partially or totally block the blood flow in an artery causing bleeding (hemorrhage) and secondly, forms blood clot (thrombus) on the plaque’s surface. If either of these occurs and blocks the arteries, a heart attack or stroke may occur. A symptom complex of Ischemia heart disease is characterized by paroxysmal attack of chest pain which is referred to as angina pectoris.
In cardiac-related problems, the affected arteries get hardened as a result of accumulation of fatty substances inside the lumen or because of formation of plaque as a result of hemorrhage. As the disease gets progress, the arteries get constricted and as such the flow behavior in the stenosed artery becomes quite different with that in the normal arteries. Also, stresses and resistance to flow parameters such as the velocity, flow rate, pressure drop will aid in formulating solutions to such problems. Hence, fluid mechanics aspect of the arterial stenoses needs to be given a considerable attention now than before.
In this work we model blood flow in a stenose tube (artery) and assumed that the flow obeys Newtonian hypothesis, in which the flow variables are computed using basic Navier Stokes equations of motion, the continuity equation where by the solution is obtained using Poisseulle’s equation.

There is a lot of havoc during blood flow due to the deposition of fatty substances in the human blood vessels. The accumulation of these substances leads to what we called STENOSIS which causes resistance to blood flow leading to most of the heart related diseases we encounter today such as     hypertension, stroke, ischemia, atherosclerosis, angina pectoris among others.

The blood flow in human arteries and veins can best be represented using fluid flow equation of motions. To properly analyze these problems for better understanding, the stability of the factors governing th flow such as velocity distribution, viscosity of blood, pressure drop, the resistance to flow, the diameter of the artery through which the blood flows  as well as the length of the  artery  under study needs to be taken into consideration.
The aim of this research is to make use of mathematical equations to show the effects of excess fats as a factor affecting blood flow, what really happened when there is a constriction (a reduction in the diameter of the artery) and then compare the behavior of blood flow in a healthy artery with that of a constricted artery.

To achieve this, we make use of the following set of objectives;
Ø   To describe the flow of blood in the artery using fluid flow equations of motion such as continuity equation and the Navier-Stokes equations.
Ø   Picking a portion of the artery as a control volume.
Ø   The Poisseuille equation of blood flow to analyze the result.

This study gives an insight on how the reduction of artery diameter affects the blood flow, hence help in early detection for necessary action. It also gives an avenue for both students of physics and mathematical sciences with little knowledge on human physiology to further research on arterial blood flow.   

The scope of this research covers blood flow through human arteries, while Navier-Stokes equations, continuity equation as well as Poisseuile equation is used in analyzing the flow.  The study is limited to mammalian physiology, in particular human beings.


1.5.1 BLOOD:
Blood is made up of a suspension of particles in a solution of proteins and electrolytes called plasma. Erythrocytes, leucocytes and platelets are the main constituents of blood. The erythrocytes or red blood cells (RBCS) are more than a thousand times numerous than the leucocytes or white blood cells (WBCS) and much more larger than the platelets.

The hematocrit (percentage of the blood volume that is made up of the red blood cells) is the major determinant of of blood viscosity and also constitutes blood flow. Just as an engine doesn’t function without fuels, lubricants and other forms of fluids, so also living things can’t function without fluids in their systems, specifically human beings.

Generally, blood is essential for life (self state). The blood transports substances to the various part of the body where they are needed. An increase in cells leads to an increase in the volume of blood i.e. the more the cells in the body, the more the function which implies a greater viscosity. The coefficient of viscosity of man at room temperature i.e. 37 degree Celsius(
C) is 0.0027N-S/ .

1.5.2   FATS:
This is a natural substance occurring in animal bodies especially when deposited as a layer under the skin or around certain organs. The food we eat contains nutrients that provides energy and other substances the body needs. Most of the nutrients fall into three major groups i.e. proteins, fats and carbohydrates. Fats serve as a fuel source since they are the major storage form of energy in the body.

1.5.3 TYPES OF FATS                                                                                                    
Saturated Fats:
These are fats that are solid at room temperature and as such, they are not good for the body particularly people who does not engage in regular exercise. Because of their nature, they pose serious problems during blood flow. Saturated fats are found in animal products such as butter, cheese, whole milk, ice cream and fatty meals. They can also be found in some vegetable oils like coconut oil, palm oil and palm kernel.

ii. Unsaturated Fats:
These are fats that help to lower blood cholesterol if used in place of saturated fats. Examples are those fats that are extracted from vegetables with the exception of coconut, palm and palm kernel.

There are two types of unsaturated fats;                                                                                                        Monounsaturated fats which include olive and canola oils                                                                Polyunsaturated oils which include fish, safflower, sunflower, corn and soyabeans.

iii. Trans fatty acids:
These are fats formed when vegetable oils are hardened. They are found in fried foods, commercial baked foods liedunnuts, crackers, cookies and margarines.

iv. Hydrogenated and partially hydrogenated fats:
Hydrogenated fats refers to oils that have  become hardened such as hard butter and margarine, while partially hydrogenated means when the oils are partially hardened. Foods made with hydrogenated fats should be avoided because they contain high levels of trans fatty acids which are linked to heart diseases.


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